Take $n=1$ and $m=p^c$ .

MarkBcc168 wrote: Take $n=1$ and $m=p^c$ . In this case $\frac{\textup{lcm}(n+1,n+2,\cdots,m)}{\textup{lcm}(n,n+1,\cdots,m-1)}=p$ doesn't satisfy the condition.

take $m=p^{2c-1}$, $n=p^{2c-1}-p^{c-1}$

take $m=p^{c+1}$and$n=m-p$

zsgvivo wrote: take $m=p^{c+1}$and$n=m-p$ Can you explain why you choose those number? I cant understand

zsgvivo wrote: take $m=p^{c+1}$and$n=m-p$ Why this satisfied the requirement, i see that the ratio is less then $p$